Decision-making by an authority under influence

Economics Letters 0165-1765/93/$06.00

43 (1993) 35-39 0 1993 Elsevier

35 Science

Publishers

Decision-making influence

B.V. All rights

reserved

by an authority

under

Torben Tranzs * Institute of Economics, Received Accepted

University of Copenhagen,

Studiestrrede 6, DK-1455

Copenhagen

K., Denmark

30 March 1993 29 June 1993

Abstract A characteristic set of decision problems is given when an authority has to choose between a finite number of alternatives and is being influenced by a number of parties. For instance, the location of a firm when local authorities are competing for being the host, various types of lobbyism, or when a manager is being influenced by deputies. This paper argues that these problems can be analyzed as decision-making in a hierarchy using a common-agency framework. The main result is that even when the influential parties act simultaneously and the strategy sets are discrete, solutions to such decision problems as pure-strategy equilibria exist.

1. Introduction The common-agency problem analyzed and applied by Bernheim and Whinston (1985, 1986) represents a characteristic strategic conflict embodied in a variety of economic institutions: market interaction as well as hierarchies within organizations. The set of problems we analyze is a subset: consider an authority about to make a decision. The authority faces a finite number of indivisible alternatives; the authority’s preferences are being influenced by a number of parties, but the payoffs from a given alternative are influenced by one party only (or none). Our intent is to highlight the relevance of the common-agency framework for this characteristic set of problems and to report on some results that facilitate applications. a government wishes to consider Within the set of problems under focus is lobbyism: suggestions from several interest groups before deciding upon a new project. The suggestions have to be alternative projects and the government intends to implement exactly one of them, e.g. a pension scheme, where and how to locate a river crossing, etc. The case when several local governments compete for being the host of an important activity is covered as well: if the losing communities are unaffected by the conditions provided for by the winning community, the principal-agent analysis of HolmstrGm (1979) and Grossman and Hart (1983) applies. If there are externalities, a common-agency framework has to be considered. And if only the externalities induced by the activity when installed are considered (under the conditions provided for by the winning local government), the framework below applies. Our model also applies to the decision-making of hierarchies. Consider, for instance, a decision * I wish to thank suggestions.

Vijay

Krishna

and

Birgitte

Sloth,

and

in particular

Birgit

Grodal

for comments

and

very

helpful

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problem within a firm: a number of deputies give their suggestions to the director who, after hearing all the suggestions, decides which one is to be implemented. Only one decision can be implemented, and when the director has made her choice - i.e. identified the deputy who gave the best suggestion from the director’s point of view - the game ends, and everybody receives the payoff which goes with the implemented suggestion. Ex post, these payoffs are independent of the suggestions from those of the deputies who did not succeed in persuading the director, and our result applies. Recent work on common-agency problems includes Baron (1985), who studies non-cooperative regulation using a common-agency type model. This is elaborated in a study by Spiller (1990); later Stole (1991) studied mechanism design under a common agency. Existence for the whole set of common-agency games requires that the problem, by assumption, is prepared for the use of a fixed point argument, as in the case in Bernheim and Whinston (1986). As for the subset of common-agency games considered here, we can prove the existence of pure-strategy subgame perfect Nash equilibria (without the assumption of convex strategy sets). Moreover, generically games of this set are unique weak dominance solvable. In section 2 we set up the basic game and definitions, and in section 3 we characterize the relevant class of games and report the results.

2. The basic game The basic setup is a common-agency game formulated in the simplest way, just capturing the particular strategic conflict involved. Let r = (N, T, (Ql)rEM, P, f) be a common-agency game; N = M U {a} is the set of players, where M is the set of principals and a is the agent (#M = m). According to the game tree, T, the game is played in two steps. First, all principals choose a strategy simultaneously, which is observed by the agent and all the principals. A strategy of player i is qi E Q,, for all i E M. Second, the agent chooses an action a E P, and the game ends. Notice that the choice set of the agent is not a function of the strategy profile chosen by the principals. Hence, a strategy of the agent is a function p : Q-P, where-Q = niEM Qi. Let p be the set of all such functions; a strategy profile is then a pair (q, p) E Q x P. All strategy sets are assumed to be finite. Furthermore, let q/q: be the strategy profile where everybody plays q, except principal i who plays q:. Payoff functions are defined directly on the set of strategy profiles, i : Q X p-+ R. The game is of almost perfect information, i.e. it is a multistage game with observed actions. Therefore it is sufficient to look at subgame perfect Nash equilibria (SPE). Analytically we will concentrate on the first simultaneous step of the game by looking at truncations like the following: for every strategy profile of the principals the payoffs corresponding to a best reply of the agent are substituted for the corresponding subgame (of depth one). If the agent has a unique best reply for all q E Q, there is one truncated game only, otherwise there are more. The set of truncated games is called the principals’ contest, G = {(M, (Qi),tM, f’)}fSEF, where F is the set of all functions f’: Q + iw”, where for all i E M, f’(q) E {f(q, p)l p E argmaxp,,p f,(q, p’)}. Each element of G is a well-defined normal form game Gr,. However, since the principals’ contest is settled before the agent moves, we wish to have a solution concept in terms of the whole of G and not just for the elements one by one. Definition. A strategy profile q* is an equilibrium of the principals’ contest does not existfIEF and q, such thatf:l (q*/q;)>fj(q’), for allf”EF. Notice

that if the agent has a unique

G if for all i E M there

best reply for all q E Q (G is a singleton)

an equilibrium

of

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G is simply a Nash equilibrium. This suggests that by having discrete strategy sets we will get problems with existence of equilibria in pure strategies. However, the set of common-agency games in which we are interested imposes enough restrictions on the game to make pure-strategy equilibria exist.

3. Decision-making

by a hierarchy

A certain class of common-agency problems is characterized by the following. For any given strategy choice of the agent there is one or none of the principals whose strategy choice matters for the payoffs. In other words, all perturbations of strategy profiles where the strategy of the agent and of this particular principal (if one exist) is fixed, yield the same payoff vector. This feature is captured by Assumption I below, where p is the subset of p containing all constant functions. Assumption

I. VbE

p, 3iEM,

Consider next all p the argmax of f,( 4, p) The existence of Assumption I, we can G; then the existence

Vq, q’EQ:

f,(q/q,,p)

=f,(q’/qi,p),

for all HEN.

E p, where &(q, p) =J(q’, ~7) for all q, q’ E Q, and all i EN, and let p0 be over all such p, where @ is just any strategy profile. pure-strategy equilibria facilitates applications, and fortunately, under prove the existence of pure-strategy equilibria for the principals’ contests of pure-strategy SPE in common-agency games, r, follows suit.

Proposition 1. If the common-agency game r satisfies Assumption I, then the principals’ contest G has at least one pure-strategy equilibrium. Proof. Consider the principals’ Then for all i E M define

contest

G of any common-agency

game r satisfying

Assumption

I.

First we assume that p = UltM p, , i.e. we ignore PO for now, but will return to the matter at the end of the proof. Observe that for J? E pi, and all qi E Qj , &( q, @) is independent of ( qk)k+i for all players j EN. Hence, we can, for all i EM, where p, is non-empty, define h,(q,) = maxbEp, f,(el profile. Furtherw h ere 4 is just an arbitrary %9), and R,(q,)={pEP,If,(~iq,,p)=h,(q,)>, more, define for all i E M, where P, is non-empty, Ql = {qi E Q, 1q, E argmin h,(q,)}, i.e. if the agent makes a best reply to the strategy of player i, then Q, is the set of strategies that minimizes the payoff to the agent. Of course, Qi need not be a singleton. Therefore, let 9; be an arbitrary clement of Qi that maximizes h( G/q:, p) over the set {(q,‘, j) Eel x pi 1p E R,( 4:)). If for some player Pi is empty, we select just an arbitrary element of Qi as gi. For these players we let Qi =q,. By naming the strategies we can of course assume that Q, n Q, = 0 for i f j. Thus, let the b e ranked according to the payoff to player a, i.e. all q, E .9 are elements in 9 = U,,,(Q,\{LJ,}) ranged according to h,(q,): the one in the ranking most preferred we label q’; the index for the player is suppressed, but she is named ~(1). The next in line is number 2, q*, and so forth. If two elements of 9 are equally preferred they are just put in arbitrary order. Finally, let the least preferred in the ranking, which is at least as good as q”““, be labelled qK, where _q”“” is an arbitrary element of (4, l4 1h,(q,) 2 h,(s)Vs q} .

l

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Owing to Assumption I, if player ~(1) plays ql, and the agent makes a best response, payoffs are independent of the strategy played by the players j # ~(1). Therefore, for all q = (q/ql) E Q there does not exist j f b(l), q;, and f’ E F such that f;(qlq;) >fj(q), for all f” E F. This statement is generalized below as Sl. We then define a sequence of contest recursively from G: first, the strategy q1 E 22 is removed from the strategy set of the principal to whom it belongs, ~(1). The contest obtained by removing q1 from Q,,,,, and thereby from G, is called G_, with the set of strategy profiles Q-‘. Next, q* is removed from G- 1, and the one remaining is contest G-, with the set of strategy profiles Q-‘, and so forth down to the restricted contest G_,, where the player holding q”“” has only this strategy left. The whole of q is of course contained in G_,. Thus the sequence of contest G_,, G -(K-i), . . . > G -1, G has been defined, and we can generalize the statement from above. Sl:

for all q = (q/qh) E Q- w’) there fl(qlqi) >fj(q), for all f” E F.

does not exist j # b(h), q; E QJCh-‘),

and f’ E F such that

Again Sl follows directly from Assumption I. Finally, we will prove existence of equilibria in G by induction after k in the sequence of q is an equilibrium of G_,, hence we contest G_(,_,,,, . . . , G_cK-kj,. . . , G_,, G. Obviously, have existence for k = 0. Then assume that for k 2 0 there exists an equilibrium, which we write E(G-(,-,,) #PI, and let q’ E E(G_(,_,,). If q’j?JE(G~C,_,,+,,,), then for player L(K - k) with strategy

qKmk, f’ E F exists such that f:(K_k)(q’lqK-k)

>f:CK_kj(q’)

,

for all f” E F .

But then (q’/qKek) E E(G_ CK_ Ck+ljj) because of Sl. Consequently, E(G_(K_(k+l))) #0. Thus, existence for k 2 0 implies existence for k + 1, which together with E(G_,) # 0 concludes the proof for the case where p = UiEMpi. For the general case we also have to take p0 into account. Assume that LjfqKmk is an equilibrium of G if we ignore PO as above, and let f,(G, &) ?f,((G/qKpk, p’)), p’ E R,(qKek). If K-k is not an equilibrium of G given that p = UiEM 41s f’, U h%~~thenf,(@, PO)f,(((i/ql, p”)), ~7”E R,(q’). Thus by repeating the induction argument from above this implies that a profile of G given that E’ = UiEM pi U {PO} (where of course 414 Kp(k+r) exists, which is an equilibrium k+t-rK-1). 0 Let_the elements of the agent’s action set P be arbitrarily numbered, and consider a best reply p* E P where, whenever in a subgame the agent is indifferent, she chooses the action with the lowest number. Given p* the principals’ contest is a singleton, then an equilibrium is just a Nash equilibrium, and by Proposition 1 we have existence in pure strategies. Hence, we have proved: Theorem 1. If the common-agency pure-strategy SPE.

game r satisfies Assumption

I, then it possesses at least one

Unfortunately we cannot hope for uniqueness. Nevertheless, if we restrict attention common-agency games we can provide a kind of uniqueness by iterative elimination dominated strategies. By generic games we mean the following:

to generic of weakly

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P) #JXq’,

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any

two

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Letters 43 (1993) 35-39

(q, p)

and

(q’, p’):

if (q, p) Z (q’, p’)

then

P’) for all i E iv.

With this assumption the principals’ contest equilibrium of G is simply a Nash equilibrium.

G is a singleton

because

so is F, and hence

an

Theorem 2. If the common-agency game r satisfies Assumptions I and II, then the principals’ contest G is weak dominance solvable, and the payoffs given by the solution are independent of the order of elimination. 1, and let F = {f ‘}. Then Proof. Let p0, 2, qh, and Qmh be defined as in the proof of Proposition from Assumptions I and II, and the way G is formed, it follows that for all q, q’ E QpChp’) : fi(q/ under Assumptions I and II, the principals’ contest G can f or all i EN. Therefore, qh) =fKq’/qh) be viewed as the normal form representation of the following extensive form game, r,: initially the agent chooses between ‘PO and ‘not p0’. If ‘PO’ is chosen, the game terminates with the payoffs the principal holding f(g, PO) (4 is just any profile), and if ‘not PO is chosen, then player I, strategy q’ , chooses one of the following two actions: ‘q” or ‘not q”. If ‘q” is chosen, the game terminates with the payoffs f’(i/q’), and if ‘not q” is chosen, then player 42) has to choose between ‘q2’ and ‘not q2’, and again, if ‘q2’ is chosen, the game terminates with the payoffs f’(41(q1, q’)), and if ‘not q2’ is chosen, the game continues, and player 43) has to move, etc. Principal L(K) is the last one to move. Since game r, is of perfect information and generic as assumed, game G is weak dominance solvable, and furthermore any order of elimination will yield the same payoffs [by Gale (1953)] 0 It should be remarked that the proof of Theorem 2 also implies using the following procedure. The agent can implement p0; if she asking player ~(1) whether she wishes to suggest q’. If so, q’ is continues by asking player ~(2) whether he wishes to suggest q2,

that we can find an equilibrium chooses not to, she proceeds by implemented; if not, the agent and so forth.

References Baron, D., 1985, Non-cooperative regulation of a non-localized externality, Rand Journal of Economics 16, 553-568. Bernheim, D. and M. Whinston, 1985, Common marketing agency as a device for facilitating collusion, Rand Journal of Economics 16, 269-281. Bernheim, D. and M. Whinston, 1986, Common agency, Econometrica 54, 923-942. Gale, D. 1953, A theory of n-person games with perfect information, Proceedings of the National Academy of Science, USA, 39, 396-501. Grossman, S.J. and O.D. Hart, 1983, An analysis of the principal-agent problem, Econometrica 51, 7-45. Holmstrom. B., 1979, Moral hazard and observability, Bell Journal of Economics 10, 74-91. Spiller, P., 1990, Politicians, interest groups, and regulators: A multiple-principals agency theory of regulation (or Let Them Be Bribed), Journal of Law and Economics 33, 65-101. Stole, L., 1991, Mechanism design under common agency. Working Paper No. 21-91, The Foerder Institute for Economic Research, Tel-Aviv University.